JOURNAL OF APPLIED PHYSICS
VOLUME 95, NUMBER 10
15 MAY 2004
Energy deposition model for low-energy electrons „10–10 000 eV… in air
A. Roldán and J. M. Pérez
Centro de Investigaciones Energéticas Medioambientales y Tecnológicas, Avenida Complutense 22,
28040 Madrid, Spain
A. Williart
Departamento de Fı́sica de los Materiales, Universidad Nacional de Educación a Distancia,
Senda del Rey 9, 28040 Madrid, Spain
F. Blanco
Departamento de Fı́sica Atómica, Molecular y Nuclear, Universidad Complutense de Madrid,
Avenida Complutense s.n., 28040 Madrid, Spain
G. Garcı́aa)
Instituto de Matemáticas y Fı́sica Fundamental, Consejo Superior de Investigaciones Cientı́ficas,
Serrano 121, 28006 Madrid, Spain
共Received 9 September 2003; accepted 23 February 2004兲
An energy deposition model for electrons in air that can be useful in microdosimetric applications
is presented in this study. The model is based on a Monte Carlo simulation of the single electron
scattering processes that can take place with the molecular constituents of the air in the energy range
10–10 000 eV. The input parameters for this procedure have been the electron scattering cross
sections, both differential and integral. These parameters were calculated using a model potential
method which describes the electron scattering with the molecular constituent of air. The reliability
of the calculated integral cross section values has been evaluated by comparison with direct total
electron scattering cross-section measurements performed by us in a transmission beam experiment.
Experimental energy loss spectra for electrons in air have been used as probability distribution
functions to define the electron energy loss in single collision events. The resulting model has been
applied to simulate the electron transport through a gas cell containing air at different pressures and
the results have been compared with those observed in the experiments. Finally, as an example of
its applicability to dosimetric issues, the energy deposition of 10 000 eV by means of successive
collisions in a free air chamber has been simulated. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1704864兴
I. INTRODUCTION
material and usually fills the ionization chambers which are
the basis of some radiation detectors. Monte Carlo methods
are commonly used to simulate the electron interaction and
transport through the matter. Most of them use, as input parameters, cross section data calculated in the framework of
the first Born approximation.1 However, we have shown in
previous studies2,3 that, for energies below 10 000 eV, this
approximation overestimates the total electron scattering
cross sections from several molecular targets, as those constituting air, reaching discrepancies of the order of 40% for
energies below 1000 eV. Therefore models based on a more
precise description of the electron scattering processes are
needed at these energies.
These considerations motivated this work. The aim of
this study is to obtain a realistic energy deposition model for
electrons in air, based on a Monte Carlo simulation of the
particle tracks and the energy losses derived from single
electron scattering events with the main molecular constituent of air, namely N2 and O2 . The simulation program was
based on the Geant44 code with significant modifications demanded by the physics of this problem. The main input parameters are the electron interaction cross sections, both differential and integral, corresponding to the most probable
processes occurring in the considered energy range 共10–
Important radiological and dosimetric applications require detailed energy deposition models of electrons interacting with matter, not only in terms of the total absorbed dose
but giving the energy distribution along the particle tracks.
Energetic particles 共photon, electron or ion兲 passing through
matter produce secondary electrons that are mainly responsible for the energy transfer by means of successive collisions with the constituent atoms or molecules. Therefore
these models should consider secondary electron interactions, at molecular level, over a wide energy range. In principle, energies from the incident particle energy down to the
final step of the energy degradation procedure should be considered. However, the great diversity of possible processes in
this range makes difficult a complete treatment of the problem, being necessary reasonable approximations supported
by experiments. Among the wide variety of targets of interest
in these applications, air is one of the most suitable, as long
as this gas is commonly adopted as a reference radiological
a兲
Author to whom correspondence should be addressed. Instituto de Matemáticas y Fı́sica Fundamental, Consejo Superior de Investigaciones Cientı́ficas, Serrano 121, 28006 Madrid, Spain; electronic mail:
[email protected]
0021-8979/2004/95(10)/5865/6/$22.00
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© 2004 American Institute of Physics
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5866
Roldan et al.
J. Appl. Phys., Vol. 95, No. 10, 15 May 2004
FIG. 1. Differential elastic cross sections for electron scattering from 共a兲
N2 , 共b兲 O2 .—, present calculation; 䉱,
DuBois and Rudd 共see Ref. 12兲; 〫,
Jansen et al. 共see Ref. 13兲; 䉲, Bromberg 共see Ref. 14兲; 䉭, Iga et al. 共see
Ref. 15兲.
10 000 eV兲. The required electron scattering cross-sections
data have been calculated for N2 and O2 by using an improved optical potential procedure.5 The air composition assumed for these calculations, as well as the reliability of the
calculation procedure have been checked by comparison
with accurate measurements 共within 3%兲 of the total electron
scattering cross sections from air performed in a transmission
beam experiment. Normalized energy loss distribution functions have been derived from the experimental energy loss
spectra obtained with the same experimental system.
The present energy deposition model has been applied to
simulate the electron beam transport through a gas cell containing air at different pressures. The energy and intensity
distributions of the simulated transmitted beam will be discussed by comparison with the experimental results. Finally,
the applicability of this energy deposition model to dosimetric issues has been shown by simulating the particle tracks
and the energy degradation process of 10 000 eV electrons
passing through a chamber filled with air at atmospheric
pressure.
II. CROSS SECTION DETERMINATION PROCEDURE
AND ENERGY LOSS SPECTRA
A. Differential and integral electron scattering cross
section calculations
The principles of the optical potential procedure followed to calculate differential and integral electron scattering
cross sections from molecules have been described in previous papers.6,7 Basically, we have considered that the incident
electron energy is appropriate for the application of the independent atom model.8 The electron–atom interaction is
then represented by an approximate optical potential given
by
V opt共 r 兲 ⫽V 共 r 兲 ⫹iV 共 r 兲
a
⫽V s 共 r 兲 ⫹V e 共 r 兲 ⫹V p 共 r 兲 ⫹iV a 共 r 兲 ,
共1兲
where the real part V(r) is the effective atomic potential
which includes three terms: the static potential calculated
from the charge density derived from a relativistic Hartree–
Fock procedure9 V s (r), the exchange potential V e (r) as defined by Riley and Truhlar10 and the polarization potential
V p (r) in the form given by Zhan et al.11 The imaginary part
V a (r) is the absorption potential derived from an ab initio
procedure based on the quasifree electron model,6,7 including
all the improvements that we have proposed recently.5 Following the method described in Ref. 5, the complex partial
wave phase shifts derived from the above potential allowed
to determine the differential elastic cross section
d ␴ el(E)/d⍀ and, applying the optical theorem,8 the total absorption or integral inelastic cross sections ␴ inel(E). The integral elastic cross sections ␴ el(E) were determined by integrating the corresponding differential values for a given
energy. Finally, molecular cross section data have been derived from those of the constituent atoms following the addition procedure described in Ref. 5.
The calculated differential elastic cross sections for electron scattering from N2 and O2 are shown in Fig. 1 for some
incident energies 共50, 500, 1000, 3000, and 10 000 eV兲, together with representative experimental data.12–15 Taking
into account that experimental differential cross section values usually have large error limits 共above 20%兲 because of
the normalization procedures needed to obtain absolute values, there is a reasonable agreement between theory and experiment for both molecules.
Integral cross-section values for electron interaction in
dry air have been determined from its molecular constituents
by assuming a target composition of 78.09% N2 and 21.19%
O2 . Results for the integral elastic, integral inelastic, and
total electron scattering cross section from air are shown in
Fig. 2 for energies ranging from 10 to 10 000 eV.
As will be described in the next section, the consistency
of these results has been checked by comparison with direct
measurements of the electron beam attenuation in air for selected energies.
B. Total electron scattering cross section and energy
loss measurements
Total electron scattering cross section from air have been
directly measured in the energy range 1000–5000 eV by using a transmission beam technique.16 The experimental setup
is shown in Fig. 3. Both the system and the experimental
procedure have been described elsewhere17,18 and will only
be briefly reported here. The primary electron beam, produced by an emitting filament, was collimated into a 1 mm
diameter beam and deflected by a combination of electro-
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J. Appl. Phys., Vol. 95, No. 10, 15 May 2004
FIG. 2. Integral electron scattering cross sections from air. Present
calculations:—, total electron scattering cross section;---, integral elastic
cross sections;-•-•-, integral inelastic cross sections. Present measurements:
䉱, total electron scattering cross sections.
static plates and a transverse magnetic field. Typical operating currents were between 10⫺13 to 10⫺14 A with an energy
resolution of 500 meV. The collision chamber was defined by
two apertures each of 1 mm diameter separated by a length
of 100 mm. Electrons emerging from the collision chamber
were deflected by an electrostatic plates system to select the
angle of analysis. The transmitted electrons were energy analyzed by an electrostatic hemispherical spectrometer combined with a retarding field. Under these conditions a constant energy resolution of about 0.8 eV was obtained over the
whole energy range considered here. The acceptance angle of
the analyzer was in the order of 10⫺5 sr. Electrons were
detected at the exit of the energy analyzer by a two stage
microchannel plate 共MCP兲 electron multiplier operating in
pulse counting mode. The gas pressure in the chamber was
measured within 1% by an absolute capacitance manometer
共MKS Baratron 127 A兲. The ultimate pressure in the region
of the energy analyzer and electron detector was lower than
10⫺6 Torr. The total cross sections were derived from the
attenuation of the transmitted electron beam when the target
pressure was varied from 1 to 20 mTorr. In these conditions,
we have shown2 that multiscattering processes are negligible
and that the error contribution to the attenuation measure-
FIG. 3. Experimental setup used for total electron scattering cross sections
and energy loss measurements.
Roldan et al.
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FIG. 4. Experimental energy loss spectrum at 5° for 2000 eV incident electrons and 20 mTorr of air in the collision chamber.
ments due to forward scattered electrons is less than 1%. As
a combination of the partial error sources, a total experimental error of 3% can be assumed for the present total crosssection measurements.16 Experimental results for total electron scattering cross sections in air are shown in Fig. 2
together with those calculated with the above procedure. As
this figure shows, for energies above 1000 eV there is a
general agreement, within 5%, between the present theoretical and experimental data. We can expect a similar behavior
at lower energies while the above method to calculate electron scattering from molecules is applicable, above 50 eV.
Therefore, for the computing procedure proposed in this paper, the calculated integral cross-section data will be used by
assuming, for this energy range, an error limit of about 7%.
Below 50 eV the results of this study should be considered
only qualitatively.
The electron energy loss spectra have been obtained with
the same experimental system by sweeping the retarding
field at the entrance of the energy analyzer with a voltage
ramp generator synchronized with a multichannel system
controlled by a computer. In this case, some changes in the
scattering geometry have been introduced to improve the angular definition. The scattering chamber length was reduced
to 10 mm while the exit aperture diameter was enlarged to 2
mm. The angle of analysis was selected by deflecting the
transmitted beam with the electrostatic parallel plate system,
covering an angular range from 0 to 15 degrees. For the
energies considered in this experiment the differential electron scattering cross sections are strongly peaked on the forward direction, being the ratio between the intensity of electrons scattered at higher angles than 15° and that of the
forward scattered electrons less than 0.1 for energies above
500 eV. A typical energy loss spectrum of electrons scattered
at 5° for 2000 eV incident energy and 20 mTorr of air in the
collision chamber is shown in Fig. 4. Energy spectra were
recorded at different incident energies ranging from and different scattering angles between 0° and 15°. We found that
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5868
J. Appl. Phys., Vol. 95, No. 10, 15 May 2004
FIG. 5. Energy loss distribution function, from 0 to 80 eV, used to sample
the energy deposition of inelastic collisions.
the scattered signal decreased as the scattering angle increased, but maintained the relative intensity distribution of
the inelastic peaks fairly constant, within 15%, for energies
above 1000 eV. We have therefore assumed a common energy loss spectrum for the whole energy range considered
here. This is a rough estimation which gives the main contribution to the uncertainty quoted to this simulation procedure. The averaged experimental energy loss spectrum defined a common probability distribution function for all
energies considered which was introduced in the simulation
to decide the energy lost by the electrons when inelastic collisions take place. This energy loss distribution function
共from 0 to 80 eV兲 is shown in Fig. 5.
III. MONTE CARLO SIMULATION
A. Simulation model
In order to implement a Monte Carlo simulation based
on original cross-section data, two possibilities can be considered: 共a兲 generating a totally new simulation code and 共b兲
modifying an existing code, replacing some theoretical approximations used to describe electron interactions by input
databases derived from experimental cross-section results.
Since a Monte Carlo simulation involves many aspects, apart
from the physics of interactions, we have concluded that the
most efficient way to implement an energy deposition model
at molecular level is by using reliable existing codes but
introducing the degree of detail required by the mentioned
applications. Thus, some advantages of significant aspects
such as generation of primaries, geometry, visualization,
tracking, etc., can be taken from them.
In principle, any of the highly developed codes available
on the market could be used for our purpose. However, one
of the largest developed codes for radiation-matter interaction simulation is Geant4.4 More than a single code, it is a
full kit of simulation tools, the result of a vast work of more
that 100 scientists and engineers from research laboratories
worldwide. As an object-oriented program, the implementa-
Roldan et al.
tion of the physics to be used is open. Specifically, the
Geant4 code gives the opportunity of using databases and
processes different to the implemented ones, allowing new
physics be tailored to the standard procedures to simulate
particles and materials. Thus, the Geant4 code can naturally
be used for our purposes of implementing new electron interaction models, taking advantage of its already tested tools
for geometry modeling, material and particle definitions,
Monte Carlo tracking of events, collection of data about interaction positions and deposited charge in sensitive materials, etc. In other words, we simply implement new physical
processes in a fully operative simulation platform, in order to
include low energy 共below 10 000 eV兲 electron interaction
processes according to our calculated and experimental results.
The electron elastic and inelastic interaction processes
have been introduced in the code as a new program. Both,
experimental and calculated cross-section data are used by
this program as input parameters. Standard Monte Carlo
techniques applied to the distribution functions generated
from the input data decide the trajectories and energy deposition patterns of electrons in the target chamber.
Once these programs are included, electrons in the code
are linked only to these effects. The standard processes in
Geant4 for electrons are not considered. Thus, the physics
applicable when an electron is tracked relies only on the two
new processes.
The mean free path of electrons in the chamber is defined by the total electron scattering cross sections. When a
single collision takes place, the program decides whether it is
elastic or inelastic by pondering the related cross-section
data. In a single elastic collision no energy loss of the electron is considered, but there is a change of direction given by
the angular probability derived from the differential elastic
cross sections computed as described above. In order to decide the angle of the emerging electron elastically scattered,
the energy range considered in this study has been divided
into nine bins 共0–150, 150–300, 300–500, 500–700, 700–
900, 900–1500, 1500–2500, 2500–3500, and 3500–10 000
eV兲 each one being represented by a fixed energy value: 100,
200, 400, 600, 800, 1000, 2000, 3000, and 5000 eV, respectively. The differential elastic cross sections corresponding to
these representative energies have been used to derive the
normalized distribution functions that have been used to
sample the scattering angles. The same procedure has been
carried out to decide the scattering angle when the collision
was inelastic.
In the case of inelastic scattering, some amount of energy is deposited in the medium. This energy lost by the
incoming electron is sorted from data tables included in the
inelastic scattering class obtained from the experimental energy loss spectra. In this way, the code provides an energy
loss profile and angular deflections fully compatible with our
experimental results. So, in contrast with other models available in the literature, the molecular structure of the target is
defining here the energy deposition pattern. In this case, the
energy assigned to the scattered electron corresponds to the
electron incoming energy 共immediately before the interaction兲 minus the energy loss.
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Roldan et al.
J. Appl. Phys., Vol. 95, No. 10, 15 May 2004
FIG. 6. Transmitted electron intensity relative to the incident electron intensity as a function of the air pressure in the gas cell for 2000 eV electrons.
The energy lost by electrons along their tracks within the
gas collision chamber is stored in the code at the corresponding position. As expected, the simulated energy loss spectrum for a 2000 eV electron beam hitting the gas cell containing air at 20 mTorr is nearly coincident with the energy
loss distribution shown in Fig. 5, for a significant number of
histories, giving the same 共within 0.1%兲 averaged energy
loss per inelastic event, namely 33.93 eV.
B. Applications to energy deposition models
The code has first been applied to the study of the attenuation of an electron beam through the gas cell at different pressures between 2 and 40 mTorr. Two collimation apertures along the axis beam, 1 mm diameter each, placed at
the entrance and exit of the chamber have been assumed to
reproduce the experimental conditions. The relative intensity
of the simulated transmitted beam is plotted in Fig. 6 as a
function of the pressure in the chamber for 2000 eV incident
electron energy. As this figure shows, the transmitted intensity follows an exponential decrease versus pressure which
allows us to determine the total cross-section value for this
energy. The agreement found between the experimental total
cross section at this energy and that derived from the simulation 共4.49 and 4.29 a 20 , respectively兲 gives an idea of the
self-consistency of the simulation program developed here.
As an example of the applicability of this procedure to
dosimetric issues, we have simulated particle tracks and energy deposition of 10 000 eV electrons interacting in a gas
chamber with air at atmospheric pressure. Electrons interact
in the gas until all its energy is deposited in it. In spite of the
relatively high energy of the primary electrons, the model
reproduces the experimental evidence that each electron deposits its energy through many interactions giving the already mentioned average energy loss. A viewgraph of a representative computed energy deposition map is shown in Fig.
7. As can be seen in this figure, the energy deposition model
proposed in this work can be a useful complement for radiation detectors, giving additional information on how and
5869
FIG. 7. Energy deposition model for 1 mm diameter electron beam of
10 000 eV entering along the X axis of a gas cell containing 1 atmosphere of
air. Giving thought to the cross-section data and the energy loss spectra, the
simulation procedure decides the type of single scattering event that takes
place, as well as the amount of energy transferred to the medium in this
process. This energy is represented by the plotted points as follows: 䊉,
between 0 and 50 eV; 䊊, between 50 and 100 eV; 䊐, higher than 100 eV. In
order to clarify the graph, only one of each 10 points are plotted.
where the energy of electrons, which are generated in these
detectors as secondary particles, is deposited inside the
chamber. In order to assign a dose magnitude to the collected
charge in these detectors some approximations are customary
used, namely the assumption of local charge equilibrium at
the collecting region and the use of a mean energy required
to produce an electron ion pair. The present simulation procedure can also be very useful to check the reliability of
these approximations for a given detector configuration, as
long as precise information about interaction positions and
energy deposition is available.
IV. CONCLUSIONS
In this paper we propose a detailed energy deposition
model for low energy electrons in gases. The model is based
on a Monte Carlo simulation which uses our experimental
and theoretical electron scattering cross-section data as input
parameters, as well as the observed energy loss spectra.
Therefore, the developed program allows input parameters
based on experimental and theoretical electron scattering
data giving information on the energy distribution along the
particle tracks. Since electrons are produced as secondary
particles in most gaseous radiation detectors, preliminary applications to the energy deposition of electrons in air at atmospheric pressure have been also presented. Some restrictive approximations, such as the assumption of an electronic
equilibrium and the definition of an average energy to produce an ion pair, are generally implicit to assess the absorbed
dose in gaseous radiation detectors which are based on
charge collection. Since these simplifications tend to fail
when the energy decreases, their reliability can be checked
through this model.
In this paper we have shown the first stage of the work in
which the main features of the method have been developed,
but considerable experimental effort to include more processes and to reduce the energy steps, and therefore the energy resolution, should be made to extend the applicability of
this model to other radiological problems 共microdosimetry,
radiation damage兲. In order to reduce the uncertainties re-
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5870
lated to the energy loss spectra, different energy distribution
functions for each energy and each scattering angle should
be also considered. These aims will be the center of attention
for further experiments.
ACKNOWLEDGMENTS
This work has been partially supported by the Spanish
Programa Nacional de Promoción General del Conocimiento
共Project No. BMF 2000-0012兲 and by the research program
of the Universidad Nacional de Educación a Distancia
共Projects Nos. 2001v/proyt/04 and 20011/ininvt/37兲.
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